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Single Reference (SR)- Mean-Field Configuration Mixing within Energy Density Functional Sly4-Bender,Bertsch,Heenen, PRL (2005) E exp – E th (MeV) E exp - E th Neutron number N Sly4-Bertsch,Sabbey, Uusnakki PRC (2005) Energy scale! Neutron number N (Skyrme, Gogny) Multi- Ref. (MR)-GCM Bender et al, PRC74 (2006) 74 Kr Mean-Field Energy Correlation Energy
Citation preview
Nuclear density functional theory with a semi-contact 3-body interaction
Denis Lacroix IPN Orsay
Outline
Infinite matter Results
Energy density function and symmetry breaking
Symmetry restoration and configuration mixing: difficulties within EDF
Problems with ra terms in Beyond Mean-Field calculation.
Proposal of a new EDF based on semi-contact three-body interaction
Collaboration: M. Bender, K. Bennaceur, T. Duguet. G. Hupin
Mean-Field Theories
Configuration Mixing within Energy Density Functional
Sly4-Bender,Bertsch,Heenen, PRL (2005)
E ex
p – E
th (M
eV)
E ex
p - E
th
Sly4-Bertsch,Sabbey, Uusnakki PRC (2005)
Energy scale!300 1000 1600
20 40 60 10080 120 140 160Neutron number N
(Skyrme, Gogny, …)To make the functional predictive,
we use and abuse of symmetry breaking
Pairing Correlations
Odd-even effects
Quadrupole Correlation
-Rotational
Bands
Surface Vibration
Translation
Rotation
Particle number
…
Single Reference (SR)-Mean-Field
Configuration Mixing within Energy Density Functional
Sly4-Bender,Bertsch,Heenen, PRL (2005)
E ex
p – E
th (M
eV)
E ex
p - E
th
Sly4-Bertsch,Sabbey, Uusnakki PRC (2005)
Energy scale!300 1000 1600
20 40 60 10080 120 140 160Neutron number N
(Skyrme, Gogny)
Multi- Ref. (MR)-GCM
Bender et al, PRC74 (2006)
74KrMean-Field
Energy
0+
0+0+
2+
2+ 2+
4+
4+
6+
8+
Corr
elati
on
Ener
gy
GCM is not so easy to use in EDF
M=91939597999
M=9
M=99
Divergence
Jump
Multi- Ref. (MR)-GCM
M: number ofMesh points
Application of conf. mixing in EDFNeeds to be regularized
Practical and conceptual difficulties in Configuration Mixing within EDF (I) Lacroix, et al, PRC79 (2009), (II) Bender et al PRC79 (2009), (III) Duguet et al, PRC79 (2009).
Correction is possible
Before correction
Corrected
SIII force
with
Example: particle number projection
or
with
Projection on Good particle number
Some uncontrolled and not understood spuriouscontributions persists M. Bender (private communication)
Duguet et al, PRC79 (2009).
The EDF energy will depend on the integration contour and becomes ill defined
We can use r, r2, r3, r4, r5 but not ra !!
The specific problem of the density dependent term ra
Example: particle number projection
Transition density
Is a complex number
Is multivalued in the complex plane
Problem: ra is very useful and practical
Interacting fermions in different regimes of density
Fictitious equation of state
Low
den
sity
regi
me
Satu
ratio
n de
nsity
regi
me
At low density
Bulgac, Forbes, …
High
den
sity
regi
me
Lee-Yang formula
Unitary gas
Galitskii formula
At saturation
Strategy : restart from true interaction to mimic density dependent term
Nuclear EDF phenomenology
Effective interaction1-body, 2-body, 3-body
?
Strategy 1
Hamiltonian1-body, 2-body, 3-body
Many-body technology
Single-Ref.
Multi-Ref.
Back to the original strategy
Constraints: - start from a Hamiltonian - no ra
Finite range ?
Zero-range(requires many terms)
Back to the construction of effective HamiltonianWith finite range interaction
Goal: mimic the proper dependence of the energy around saturation
i.e.
2-body interaction
c1 and c2 are free parameters
Infinite matter case
Convenient only for
Need for at least 3-body interaction
Semi-contact 3-body interaction
Effective Hamiltonian with 2-body and 3-body
Semi-contact 3-body interaction
Jacobi coordinate(r,R)
Idea: take a zero range in ROur starting point:
Lacroix, Bennaceur, Phys. Rev. C91 (2015) and (antisymmetrization)
EDF based on semi-contact 3-body interaction
Our starting point:
Lacroix, Bennaceur, Phys. Rev. C91 (2015)
Infinite matter case
EDF based on semi-contact 3-body interaction
EDF = 2-body + 3-body (semi-contact)
zero-rangeor
finite-range
Lacroix, Bennaceur, Phys. Rev. C91 (2015).
Illustration I : Skyrme (2-body)+3-body
Properties after a global fit:
EDF based on semi-contact 3-body interaction
EDF = 2-body + 3-body (semi-contact)
zero-rangeor
finite-range
Illustration II : Gogny (2-body)+3-body
Lacroix, Bennaceur, Phys. Rev. C91 (2015).
Properties after a global fit:
Some remarks: towards application to nuclei
Complete expression
Two difficulties :
First step : simplified interaction
Neglect the exchange with the third particle
Equivalent to the result of a density dependent finite range 2-body interaction
is not on the mesh
Exchange with the third particles
Simplified 2-body interaction: some remarks
Can we still mimic a ra behavior ? Yes
2-body functional
Simplified 2-body interaction: some remarks
We still have the difficulty
A natural solution
The functional is then equivalent to the one derived from the effective density dependent 2-body interaction:
Same as F. Chappert (PhD) with a=1Chappert, Pillet, Girod, Berger, Phys. Rev. C92 (2015)
Summary
To mimic density-dependent term, we proposed a new 3-body interaction
Gives a good description on infinite systems In different spin isospin channels
M=91939597999
M=9
M=99
DivergenceJump
Combining Nuclear EDF and conf. Mean-field leads to specific problems
Several solutions have been proposed
-regularization of divergences -New EDF with 2-body density matrix
Future dev: application to finite systems
One solution is to go back to effective HamiltonianAnd keep consistency between mean-field and pairing